Vol. 15, No. 3, 1965

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ISSN: 0030-8730
L2 expansions in terms of generalized heat polynomials and of their Appell transforms

Deborah Tepper Haimo

Vol. 15 (1965), No. 3, 865–875
Abstract

The object of this paper is to characterize functions which have L2 expansions in terms of polynomial solutions Pn,ν(x,t) of the generalized heat equation

 ∂2-- 2ν-∂--        ∂-
[∂x2 + x ∂x]u(x,t) = ∂tu(x,t).
(*)

and in terms of the Appell transforms Wn,ν(x,t) of the Pn,ν(x,t). H denotes the C2 class of functions u(x,t) which, for a < t < b, satisfy (*) and for which

        ∫
∞          ′     ′
u(x,t) =  0 G (x,y;t− t)u(y,t)dμ(y),

dμ(x) = 2(1∕2)− ν[Γ (ν + 12)]−1x2ν dx,

for all t, t, a < t< t < b, the integral converging absolutely, where G(x,y;t) is the source solution of (*). The principal results are the following:

Theorem. Let u(x,t) H,  σ t < 0, and

              1   2
u(x,t)[G(x;− t)]2 ∈ L

for each fixed t σ t < 0, 0 x < . Then, for σ t < 0,

    ∫ ∞                ∑N
lNim→∞     G(x;− t)|u(x,t)−   anPn,ν(x,− t)|2dμ(x) = 0,
0                 n=0

and

∫ ∞                       ∑∞
G (x;− t)|u(x,t)|2dμ(x) =   |an|2b−n1t2n,
0                        n=0

where

              Γ (ν + 1)
bn = [24nn!]−1-----12---,
Γ (ν + 2 + n)

and

       ∫ ∞
an = bn    u(y,t)Wn,ν(y,− t)dμ(y).
0

Theorem. If u(x,t) H, 0 < t σ, and if

u(ix,t)[G(x;t)]12 ∈ L2

for each fixed t, 0 < t σ, 0 x < , then, for 0 < t σ,

     ∫                 N
lim    ∞G (x;t)|u(ix,t) − ∑  a P  (x,− t)|2 dμ(x ) = 0,
N→ ∞  0               n=0  n n,ν

and

∫ ∞             2        ∞∑     2− 12n
0  G(x;t)|u(ix,t)|dμ(x) =    |an| bn  t ,
n=0

where bn is given above and

      ∫
∞
an = bn 0 u(ix,t)Wn,ν(x,t)dμ (x).

Theorem. If u(x,t) H, 0 < σ t, and if

            1
u(x,t)[G (ix;t)]2 ∈ L2

for each fixed t, 0 < σ t, 0 x < , then, for 0 < σ t,

     ∫ ∞                N
lim      G(ix;t)|u (x,t)− ∑  anWn,ν(x,t)|2dμ(x) = 0,
N →∞  0                n=0

and

∫                         ∞
∞ G(ix;t)|u(x,t)|2dμ(x) = ∑ t−2nb−1(2t)−2ν−1|a |2,
0                       n=0     n          n

where bn is given above and

      ∫
∞
an = bn 0 u(x,t)Pn,ν(x,− t)dμ (x).

Mathematical Subject Classification
Primary: 44.30
Secondary: 33.00
Milestones
Received: 16 January 1965
Published: 1 September 1965
Authors
Deborah Tepper Haimo